8-6
The Law of Cosines
Objective
To apply the Law of Cosines
Essential Understanding
If you know the measures of two side
lengths and the measure of the included
angle (SAS), or all three side lengths (SSS),
then you can find all the other measures of
the triangle.
A farmer needs to put a pipe through a hill for irrigation.
The farmer attaches a 14.5 meter rope and an 11.2 meter
rope at each entry point of the pipe and makes a triangle.
The ends meet at a 580 angle.
What is the length of the
pipe the farmer needs?
Can you use Law of Sines?
x
No, you don’t know the angles
that are opposite the sides
Pythagorean Theorem?
Not a right triangle
11.2 m
14.5 m
580
Law of Cosines
For any triangle ABC, the Law of Cosines relates the cosine
of each angle to the side lengths of the triangle.
C
a2 = b2 + c2 − 2bccosA
a
b2
c2
=
=
a2
a2
+
+
c2
b2
b
− 2accosB
− 2abcosC
A
c
B
Using the Law of Cosines (SAS)
A
Find b to the nearest tenth.
b
10
B
b2 = a2 + c2 − 2accosB
44
Law of Cosines
b2 = 222 + 102 − 2(22)(10)cos44
b
16.35513644
b
22
16.4
Substitute
C
Using the Law of Cosines (SSS)
T
Find
V to the nearest tenth.
6.7
4.4
U
b2 = a2 + c2 − 2accosB
7.1
Law of Cosines
4.42 = 6.72 + 7.12 − 2(6.7)(7.1)cosV
Substitute
Solve for angle V.
19.36 = 44.89 + 50.41 − 95.14cosV
Substitute
V
Examples
Law of Cosines
c2 = a2 + b2 − 2abcosC
x2 = 11.22 + 14.52 − 2(11.2)(14.5)cos580
x
11.2 m
14.5 m
x2 = 163.6
x = 12.8 m
580
c2 = a2 + b2 − 2abcosC
42 = 52 + 72 − 2(5)(7)cosxo
16 = 25 + 49 − 70cosxo
5
4
xo
7
-58 = − 70cosxo
.829 = cosxo
34o = x